Using elliptic curves to construct 3D arrays

Abstract

We present a new method of constructing three dimensional periodic arrays by composing a two dimensional periodic array with a sequence of shifts consisting of a cyclic group of points on an elliptic curve over a prime field \({\mathbb {F}}_p\). For every base array B with period (c, c) and every cyclic group G of order C there are \(\phi (C)\) families, each of size \(p^2\), of 3D arrays with period (c, c, C). We illustrate our method using a Legendre array as base array. The resulting 3D arrays have period (p, p, C), peak auto-correlation value \(C(p^2-1)\), and non-peak auto-correlation and cross-correlation values of the form \(kp^2-C\) where C is the order of the group and, in the general case, \(k\le 3\). We present experimental results that show that \(k\le 2\) for a certain type of cyclic group of points in \({\mathbb {F}}_p\) when \(p<1000\). Finally, we show that the linear complexity of our constructions compare favorably with other known constructions.